K. V. Subrahmanyam (Chennai Mathematical Institute)
The group of diagonal matrices acts on the Grassmannian of k-planes of an n-dimensional vector space over the complex numbers. The problem of understanding the GIT quotient of this action is related to the classical problem of understanding the GIT-quotient of n points spanning projective k-space (under the action of the projective linear group). When k=2 this problem is well understood, but not much is known for other k. It is known that there is a minimal Schubert variety in the Grassmannian admitting semistable points (with respect to the Plucker embedding).
We undertake a study of the GIT quotients of interesting subvarieties in the Grassmannian which admit semistable points and prove some general theorems. In the particular case k=3 and n=7, we give a complete description of the GIT quotient and show some interesting geometric properties of the GIT quotient.
Although there are no immediate complexity theoretic connections to this work, an understanding of GIT quotients is nevertheless important from the perspective of GCT. We will assume no familiarity with GIT but we hope to motivate the general question, the vocabulary, and the combinatorics involved using torus quotients of the Grassmannian as our motivating example.We will give a brief introduction to GIT quotients and discuss some recent work with Kannan and Bakshi on torus quotients of certain Schubert varieties in the Grassmannian. An understanding of GIT quotients is essential to the GCT programme - however understanding torus quotients in the Grassmannian is already a non-trivial problem. While there are no complexity theoretic results in this work I hope to convey the rich combinatorics that comes into play. I will assume very little background and try to convey and motivate some important definitions.